Let $$\mathbf{a}$$ and $$\mathbf{b}$$ be two unit vectors such that $$|\mathbf{a} + \mathbf{b}| + 2|\mathbf{a} \times \mathbf{b}| = 2$$. If $$\theta \in (0, \pi)$$ is the angle between $$\hat{a}$$ and $$\hat{b}$$, then among the statements:
$$(S1) : 2|\hat{a} \times \hat{b}| = |\hat{a} - \hat{b}|$$
$$(S2)$$ : The projection of $$\hat{a}$$ on $$(\hat{a} + \hat{b})$$ is $$\frac{1}{2}$$

A

Only $$(S1)$$ is true.

B

Only $$(S2)$$ is true.

C

Both $$(S1)$$ and $$(S2)$$ are true.

D

Both $$(S1)$$ and $$(S2)$$ are false.